The gauge invariance and transversality of the higher-spin super-Cotton tensor means that its associated higher-spin Chern-Simons-type functional [1]}, [2]}
\(S^{(n)}_{\rm SCS}[H] =- \frac{{\rm i}^n}{2^{\left\lfloor {n/2}\right\rfloor +1}} \int {\rm d}^{3|2}z \, E \, H^{\alpha (n)} \mathfrak {W}_{\alpha (n)}(H) ~, \qquad E^{-1}= {\rm Ber }(E_A{}^M)\) 
